Graceful Diameter-Six Trees with Matching Deficiency at Most One

For a tree T, let def(T) = |V(T)| - 2ν(T), where ν(T) is the maximum matching number. Thus def(T) = 0 means that T has a perfect matching, while def(T) = 1 means that T has a matching covering all but one vertex. We prove that every tree of diameter six and matching deficiency at most one is graceful. The key construction is an odd near-corona lemma. If K is a nontrivial graceful tree and b is a vertex of K, then the tree obtained by adjoining one private leaf to every vertex of K except b is graceful. Its old edges realize all even differences, while an extended odd Langford sequence realizes all odd differences. An exact matching formula divides the deficiency-one diameter-six trees into four faces. Three faces are near-coronas of trees of diameter at most five. The fourth face is reduced, using strongly graceful contree lifts and the specified-zero classification for diameter at most four, to one explicit two-parameter family, which is closed by a four-edge cap. No finite search or solver result is used. AI-assisted tools disclosure: OpenAI ChatGPT and Codex were used as aids in proof exploration, literature retrieval, symbolic checking, and manuscript organization. The author assumes full responsibility for every mathematical statement, citation, and final verification.